This will be your opportunity to be the teacher. Click on “View Full Description and attachments” below for the directions and questions. Be sure to open the file that says “MATH110 Read This First” before you jump in!
Pick ONE of the problems that has not already been solved, and demonstrate its solution using either the substitution or elimination method.
The answers are at the end of the file, so don’t just give an answer—we can already see what the answers are. Don’t post an explanation unless your answer matches the correct one!
For this forum, be sure to do the following in addition to reading the full description and the attachments from the forums main page:
-In the subject line, put the question number and a brief description of the question.
-Copy the problem statement into the body of the posting.
-Define your variables.
-Write two equations based on the question.
-State the method you are using and solve the system of equations.
-Answer the question in a complete sentence.
You do not need to response to classmates’ posts for this forum.
System of Equations Sample Post
At a dog park, there were a total of 18 dogs and people combined. There were a total of 50 legs on the dogs and people. How many dogs and people were at the park?
Let d = number of dogs at the park; then 4d = total number of legs on the dogs.
Let p = number of people at the park; then 2p = total number of legs on the people.
Equation 1: The total number of dogs and people at the park is 18: d + p = 18
Equation 2: The total number of legs at the park is 50: 4d + 2p = 50
Use substitution to solve the system of equations.
Solve Equation 1 for p by subtracting d from each side: p = 18 – d
Substitute 18 – d for p in Equation 2 and then solve for d:
4d + 2(18 – d) = 50
4d + 36 – 2d = 50 distribute on the left side
2d + 36 = 50 combine like terms on the left side
2d = 14 subtract 36 from each side
d = 7 divide both sides by 2
If d = 7, then p = 18 – 7 = 11.
Check number of legs: 4(7) + 2(11) = 28 + 22 = 50.
So there were seven dogs and eleven people at the park.
Forum: If Only I Had a System…
Applications of Systems of Linear Equalities
The Problem:
When students are surveyed about what makes a good math Forum, at least half of the responses involve
“discussing how to work problems” “seeing how this math applies to real-life situations”
This Forum on applications of systems of equations addresses both of these concerns.
Unfortunately, the typical postings are far from ideal.
This is an attempt to rectify the situation. Please read this in its entirety before you
post your answer!
Pick-up games in the park vs. the NBA:
Shooting hoops in the park may be lots of fun, but it scarcely qualifies as the precision play of a well-coached team. On the one hand, you have individuals with different
approaches and different skill levels, “doing their own thing” within the general rules of the game. On the other hand you have trained individuals, using proven strategies and basing their moves on fundamentals that have been practiced until they are second
nature.
The purpose of learning algebra is to change a natural, undisciplined approach to
individual problem solving into an organized, well-rehearsed system that will work on many different problems. Just like early morning practice, this might not always be pleasant; just like Michael Jordan, if you put in the time learning how to do it correctly, you will score big-time in the end.
But my brain just doesn’t work that way. . .
Nonsense! This has nothing to do with how your brain works. This is a matter of learning to read carefully, to extract data from the given situation and to apply a
mathematical system to the data in order to obtain a desired answer. Anyone can learn to do this. It is just a matter of following the system; much like making cookies is a matter of following a recipe.
“Pick-up Game” Math
It is appalling how many responses involve plugging in numbers until it works.
“My birthday is the eleventh, so I always start with 11 and work from there.”
“The story involved both cats and dogs so I took one of the numbers, divided by 2 and then I experimented.”
“First I fire up Excel…”
“I know in real-life that hot dogs cost more than Coke, so I crossed my fingers and started with $0.50 for the Coke…”
The reason these “problem-solving” boards are moderated is so that these creative souls don’t get everyone else confused!
NBA Math
In more involved problems, where the answer might come out to be something irrational, like the square root of three, you are not likely to just randomly guess the
correct answer to plug it in. To find that kind of answer by an iterative process (plugging and adjusting; plugging and adjusting; …) would take lots of tedious work or a computer. Algebra gives you a relative painless way of achieving your objective without wearing
your pencil to the nub.
The reason that all of the homework has involved x’s and y’s and two equations, is that
we are going to solve these problems that way. Each of these problems is a story about two things, so every one of these is going to have an x and a y.
In some problems, it’s helpful to use different letters, to help keep straight what the variables stand for. For example, let L = the length of the rectangle and W = the width.
The biggest advantage to this method is that when you have found that w = 3 you are more likely to notice that you still haven’t answered the question, “W hat is the length of
the rectangle?” Here are the steps to the solution process:
Figure out from the story what those two things are.
o one of these will be x
o the other will be y
The first sentence of your solution will be “Let x = ” (or “Let L = ” )
o Unless it is your express purpose to drive your instructor right over the edge, make sure that your very first word is “Let”
The second sentence of your solution will be “Let y = ” (or “Let W = ” )
Each story gives two different relationships between the two things.
o Use one of those relationships to write your first equation.
o Use the second relationship to write the second equation.
Now demonstrate how to solve the system of two equations. You will be using
either
o substitution
o or elimination – just like in the homework.
More examples…
For this problem, I’d use substitution to solve the system of equations:
The length of a rectangle blah, blah, blah…
Let L = the length of the rectangle
… blah, blah, blah twice the width
Let W = the width of the rectangle
The length is 6 inches less than twice the width
L = 2W – 6
The perimeter of the rectangle is 56
2L + 2W =56
For this one, I’d use elimination to solve the system of equations:
Blah, blah, blah bought 2 cokes…
Let x = the price of a coke
.. blah, blah, blah 4 hot dogs
Let y = the price of a hot dog
2 cokes plus 4 hot dogs cost 8.00
2x + 4y = 8.00
3 cokes plus 2 hot dogs cost 8.00
3x + 2y = 8.00
For this one, I’d use substitution to solve the system of equations:
One number is blah, blah, blah…
Let x = the first number
…blah, blah, blah triple the second number
Let y = the second number
The first number is triple the second
x = 3y
The sum of the numbers is 24
x + y = 24
Checking your answers vs. Solving the problem
The problem: Two numbers add to give 4 and subtract to give 2. Find the numbers.
Solving the problem:
Let x = the first number
Let y = the second number
Two numbers add to give 4: x + y = 4
Two numbers subtract to give 2: x – y = 2
Our two equations are: x + y = 4 x – y = 2 Adding the equations we get
2x = 6
x = 3 The first number is 3.
x + y = 4 Substituting that answer into equation 1
3 + y = 4
y = 1 The second number is 1.
Checking the answers:
Two numbers add to give 4: 3 + 1 = 4
The two numbers subtract to give 2: 3 – 1 = 2
Do NOT demonstrate how to check the answers that are provided and call that
demonstrating how to solve the problem!
Formulas vs. Solving equations
Formulas express standard relationships between measurements of things in the real world and are probably the mathematical tools that are used most frequently in real -life situations.
Solving equations involves getting an answer to a specific problem, sometimes based on real-world data, and sometimes not. In the process of solving a problem, you may
need to apply a formula. As a member of modern society, it is assumed that you know certain common formulas such as the area of a square or the perimeter of a rectangle. If you are unsure about a formula, just Google it. Chances are excellent it will be in one of
the first few hits.
If you are still baffled:
W atch a lecture on “Applications of Systems of Equations”
W atch this video: Solve applications of systems of linear equations or inequalities
View the PowerPoint Presentations that are provided in the link from the Lesson section.
Message me if you are still confused.
Systems of Equations
1) A vendor sells hot dogs and bags of potato chips. A customer buys 4 hot dogs and 5 bags of potato chips for $12.00. Another customer buys 3 hot dogs and 4 bags of potato chips for $9.25. Find the cost of each item.
1)
2) University Theater sold 556 tickets for a play. Tickets cost $22 per adult and $12 per senior citizen. If total receipts were $8492, how many senior citizen tickets were sold?
2)
3) A tour group split into two groups when waiting in line for food at a fast food counter. The first group bought 8 slices of pizza and 4 soft drinks for $36.12. The second group bought 6 slices of pizza and 6 soft drinks for $31.74. How much does one slice of pizza cost?
3)
4) Tina Thompson scored 34 points in a recent basketball game without making any 3-point shots. She scored 23 times, making several free
throws worth 1 point each and several field goals worth two points each. How many free throws did she make? How many 2-point field goals did
she make?
4)
5) Julio has found that his new car gets 36 miles per gallon on the highway and 31 miles per gallon in the city. He recently drove 397 miles on 12 gallons of gasoline. How many miles did he drive on the highway? How many miles did he drive in the city?
5)
6) A textile company has specific dyeing and drying times for its different cloths. A roll of Cloth A requires 65 minutes of dyeing time and 50 minutes of drying time. A roll of Cloth B requires 55 minutes of dyeing time and 30 minutes of drying time. The production division allocates 2440 minutes of dyeing time and 1680 minutes of drying time for the week. How many rolls of each cloth can be dyed and dried?
6)
7) A bank teller has 54 $5 and $20 bills in her cash drawer. The value of the bills is $780. How many $5 bills are there?
7)
8) Jamil always throws loose change into a pencil holder on his desk and takes it out every two weeks. This time it is all nickels and dimes. There are 2 times as many dimes as nickels, and the value of the dimes is $1.65 more than the value of the nickels. How many nickels and dimes does Jamil have?
8)
9) A flat rectangular piece of aluminum has a perimeter of 60 inches. The length is 14 inches longer than the width. Find the width.
9)
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10) Jarod is having a problem with rabbits getting into his vegetable garden, so he decides to fence it in. The length of the garden is 8 feet more than 3 times the width. He needs 64 feet of fencing to do the job. Find the length and width of the garden.
10)
11) Two angles are supplementary if the sum of their measures is 180°. The measure of the first angle is 18° less than two times the second angle. Find the measure of each angle.
11)
12) The three angles in a triangle always add up to 180°. If one angle in a triangle is 72° and the second is 2 times the third, what are the three angles?
12)
13) An isosceles triangle is one in which two of the sides are congruent. The perimeter of an isosceles triangle is 21 mm. If the length of the congruent sides is 3 times the length of the third side, find the dimensions of the triangle.
13)
14) A chemist needs 130 milliliters of a 57% solution but has only 33% and 85% solutions available. Find how many milliliters of each that should be mixed to get the desired solution.
14)
15) Two lines that are not parallel are shown. Suppose that the measure of angle 1 is (3x + 2y)°, the measure of angle 2 is 9y°, and the measure of
angle 3 is (x + y)°. Find x and y.
15)
16) The manager of a bulk foods establishment sells a trail mix for $8 per pound and premium cashews for $15 per pound. The manager wishes to make a 35-pound trail mix-cashew mixture that will sell for $14 per
pound. How many pounds of each should be used?
16)
17) A college student earned $7300 during summer vacation working as a waiter in a popular restaurant. The student invested part of the money at 7% and the rest at 6%. If the student received a total of $458 in interest at the end of the year, how much was invested at 7%?
17)
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18) A retired couple has $160,000 to invest to obtain annual income. They want some of it invested in safe Certificates of Deposit yielding 6%. The rest they want to invest in AA bonds yielding 11% per year. How much should they invest in each to realize exactly $15,600 per year?
18)
19) A certain aircraft can fly 1330 miles with the wind in 5 hours and travel the same distance against the wind in 7 hours. What is the speed of the wind?
19)
20) Julie and Eric row their boat (at a constant speed) 40 miles downstream for 4 hours, helped by the current. Rowing at the same rate, the trip back against the current takes 10 hours. Find the rate of the current.
20)
21) Khang and Hector live 88 miles apart in southeastern Missouri. They decide to bicycle towards each other and meet somewhere in between. Hector’s rate of speed is 60% of Khang’s. They start out at the same time and meet 5 hours later. Find Hector’s rate of speed.
21)
22) Devon purchased tickets to an air show for 9 adults and 2 children. The total cost was $252. The cost of a child’s ticket was $6 less than the cost of an adult’s ticket. Find the price of an adult’s ticket and a child’s ticket.
22)
23) On a buying trip in Los Angeles, Rosaria Perez ordered 120 pieces of jewelry: a number of bracelets at $8 each and a number of necklaces at $11 each. She wrote a check for $1140 to pay for the order. How many bracelets and how many necklaces did Rosaria purchase?
23)
24) Natasha rides her bike (at a constant speed) for 4 hours, helped by a wind of 3 miles per hour. Pedaling at the same rate, the trip back against the wind takes 10 hours. Find find the total round trip distance she traveled.
24)
25) A barge takes 4 hours to move (at a constant rate) downstream for 40 miles, helped by a current of 3 miles per hour. If the barge’s engines are set at the same pace, find the time of its return trip against the current.
25)
26) Doreen and Irena plan to leave their houses at the same time, roller blade towards each other, and meet for lunch after 2 hours on the road. Doreen can maintain a speed of 2 miles per hour, which is 40% of Irena’s speed. If they meet exactly as planned, what is the distance between their houses?
26)
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27) Dmitri needs 7 liters of a 36% solution of sulfuric acid for a research project in molecular biology. He has two supplies of sulfuric acid solution: one is an unlimited supply of the 56% solution and the other an unlimited supply of the 21% solution. How many liters of each solution should Dmitri use?
27)
28) Chandra has 2 liters of a 30% solution of sodium hydroxide in a container. What is the amount and concentration of sodium hydroxide solution she must add to this in order to end up with 6 liters of 46% solution?
28)
29) Jimmy is a partner in an Internet-based coffee supplier. The company
offers gourmet coffee beans for $12 per pound and regular coffee beans for $6 per pound. Jimmy is creating a medium-price product that will
sell for $8 per pound. The first thing to go into the mixing bin was 10 pounds of the gourmet beans. How many pounds of the less expensive regular beans should be added?
29)
30) During the 1998-1999 Little League season, the Tigers played 57 games.
They lost 21 more games than they won. How many games did they win that season?
30)
31) The perimeter of a rectangle is 48 m. If the width were doubled and the length were increased by 24 m, the perimeter would be 112 m. What are the length and width of the rectangle?
31)
32) The perimeter of a triangle is 46 cm. The triangle is isosceles now, but if its base were lengthened by 4 cm and each leg were shortened by 7 cm, it would be equilateral. Find the length of the base of the original triangle.
32)
33) The side of an equilateral triangle is 8 inches shorter than the side of a square. The perimeter of the square is 46 inches more than the perimeter of the triangle. Find the length of a side of the square.
33)
34) The side of an equilateral triangle is 2 inches shorter than the side of a square. The perimeter of the square is 30 inches more than the perimeter of the triangle. Find the length of a side of the triangle.
34)
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Answer Key Testname: SYSTEMS_OF_EQUATIONS
1) $1.75 for a hot dog; $1.00 for a bag of potato chips 2) 374 senior citizen tickets 3) $3.74 per slice of pizza 4) 12 free throws, 11 field goals 5) 180 miles on the highway, 217 miles in the city 6) 24 rolls of Cloth A, 16 rolls of Cloth B 7) 20 $5 bills 8) 11 nickels and 22 dimes 9) 8 inches
10) length: 26 feet; width: 6 feet 11) first angle = 114°
second angle = 66°
12) 72°, 72°, 36° 13) 3 mm, 9 mm, 9 mm 14) 70 mL of 33%; 60 mL of 85%
15) x = 288
7 , y =
36 7
16) 5 pounds of trail mix 30 pounds of cashews
17) $2000 18) $120,000 at 11% and $40,000 at 6% 19) 38 mph 20) 3 mph 21) 6.6 mph 22) adult’s ticket: $24; child’s ticket: $18 23) 60 bracelets and 60 necklaces 24) 80 mi 25) 10 hr 26) 14 mi 27) 56% solution: 3 L; 21% solution: 4 L 28) 4 L of 54% solution 29) 20 lb 30) 18 games 31) Length: 16 m; width: 8 m 32) 8 cm 33) 22 inches 34) 22 inches
5
